From Northwest Washington to the suburbs of Fairfax and Prince George's counties, advanced algebra often appears the same from class to class: Students are expected to learn dozens of skills, including factoring trinomials, solving rational equations and graphing quadratic functions.

But behind the surface similarities, experts say, there can be wide variations in what students learn in a course seen as critical to developing a math-savvy workforce for the digital age.

Those variations reflect, in part, patchwork government policy: The District, like many states, is moving toward a mandate for all students to pass Algebra II before graduating. Maryland and Virginia are not.

On the other hand, although all three jurisdictions have raised expectations for what should be taught in the class, only Virginia requires Algebra II students to take a standardized test to show they have learned the material.

Historically, "academic standards have been all over the place," said Sandy Boyd, a vice president at Achieve in the District. The organization works with states to strengthen education standards and graduation requirements to prepare students for college or more than a dead-end job. If students can crack advanced algebra, experts say, their college chances and career prospects will be brighter and their future evenings free from rehashing the same concepts in community college.

Accordingly, 20 states and the District have made Algebra II, or an equivalent course, a must for a high school diploma, up from two states in 2005. Deborah A. Gist, the District's state superintendent of education, said the requirement, approved in 2007, was overdue.

"This is our attempt to make sure our students can stay competitive," she said.

Top students have long been expected to take advanced algebra. At Benjamin Banneker Academic High School, a college preparatory magnet school near downtown Washington, two dozen students spent a wintry afternoon multiplying matrixes. The multi-step problem, which involved manipulating three rows and three columns of numbers, would have been a quick calculator job. But Sandra Allen said she wanted to give her class some enrichment.

At the District's neighborhood high schools, though, many students struggle with rudimentary skills. On a national math test given to District eighth-graders in 2007, 34 percent scored at the basic level or better. By 2011, those students will need to pass four years of math, including Algebra II, to receive a regular diploma.

Laura Slover, a D.C. State Board of Education member who is also a vice president at Achieve, said the requirement is a first step. But to meet the goal, she said, the District must improve teacher training, support students who are lagging and synchronize instruction from preschool on.

"Raising graduation requirements does not mean that all students magically meet them," Slover said.

Some advocates of expanding access to higher math said that making Algebra II a uniform expectation is a civil rights issue because the course is widely considered an important bridge to college -- a bridge that many poor or minority students miss.

But Maryland and Virginia educators said it is important for the course to remain challenging. Officials estimate that two-thirds of the students in each state take the course now. They would like to increase that number, but not all at once.

"I want to make sure that if a student takes a course, it's really a significant course, not a watered-down version," said Ronald A. Peiffer, Maryland deputy state superintendent for academic policy.

Peiffer said that when the state made Algebra I a graduation requirement in the early 1990s, many schools began offering two versions, the traditional course and one some teachers called "baby algebra." The state tried to rectify the disparity later, mandating an end-of-course graduation test for Algebra I that students are expected to pass to receive a diploma.

Robert Bradford, a teacher at Fairmont Heights High School, a high-poverty school in Prince George's, said efforts to strengthen Algebra I through the graduation test have hurt his Algebra II classes.

Students spend so much time preparing for the test, which includes questions on data analysis and statistics, Bradford said, that they miss out on key algebraic concepts. Now he begins Algebra II at "chapter zero," covering material students should have learned but didn't. "It's a step backwards," he said.

Virginia students must pass Algebra II if they want an "advanced" diploma, which about half of graduates received last year. Other graduates do not need to take the course.

The state plans to offer another version of Algebra II with more statistics and data analysis for those pursuing technical diplomas. The course would be less theoretical, but education officials predict that it will still prepare students for college-level math.

Virginia Superintendent of Public Instruction Patricia I. Wright said that students should be given incentives to take the more challenging class but that it should not be mandated.

"We will testify to the rigor of our algebra course," she said.

Statewide, 90 percent of Virginia Algebra II students passed the end-of-course Standards of Learning exam, which means they answered correctly at least 30 of 50 questions. Thirty-six percent received an advanced rating, which means they got at least 45 answers right.

Maryland and the District are considering standardized testing to help gauge how much students are learning.

Last year, a dozen states tried an exam that they designed together, in coordination with Achieve. The test is meant to be tough, with open-ended questions as well as multiple-choice ones. In some states, only one in five students passed. A small group of Maryland students will try it out in the spring.

At Fairfax High School, Tricia Colclaser's Algebra II class moves quickly. One recent morning, after a unit on quadratic equations, she explained three approaches to graphing second-degree equations. Near the end of the lesson, a student asked, "How do you know when it's a quadratic equation?"

To help students keep up, time is left open in their schedules to seek extra instruction from teachers. And a tutoring program pairs honor students with those who want more help.

During a free period after class, junior Musa Condah, 17, often stays to finish his homework and meet with a tutor. He is a popular athlete and diligent math student.

"I'm not the best math guy, but I'm trying to pay attention," he said. "Nowadays, it's all about technology and computers. Everyone needs math."